行列の計算例題（行列式（余因子展開））

　　　　　　　　　　


例題：次の行列式を余因子展開を使って計算せよ。

\[ \left|\,\begin{array}{@{}rwr{20pt}wr{20pt}wr{20pt}@{}}2 & 2 & {-}2 & 1 \\ 1 & 2 & 0 & {-}2 \\ {-}1 & 0 & 1 & {-}2 \\ 0 & 1 & 2 & {-}2\end{array}\,\right| \]

解答

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}2 & 2 & {-}2 & 1 \\ 1 & 2 & 0 & {-}2 \\ {-}1 & 0 & 1 & {-}2 \\ 0 & 1 & 2 & {-}2\end{array}\,\right| \qquad \) （余因子展開） 第4列で展開する : [1, -2, -2, -2]

\( \qquad\quad = (-1)^{4-1}\times1\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}1 & 2 & 0 \\ {-}1 & 0 & 1 \\ 0 & 1 & 2\end{array}\,\right| + (-1)^{4-2}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}2 & 2 & {-}2 \\ {-}1 & 0 & 1 \\ 0 & 1 & 2\end{array}\,\right| + (-1)^{4-3}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}2 & 2 & {-}2 \\ 1 & 2 & 0 \\ 0 & 1 & 2\end{array}\,\right| + (-1)^{4-4}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}2 & 2 & {-}2 \\ 1 & 2 & 0 \\ {-}1 & 0 & 1\end{array}\,\right| \)

\( \qquad\quad = (-1)^{4-1}\times1\times3 + (-1)^{4-2}\times(-2)\times4 + (-1)^{4-3}\times(-2)\times2 + (-1)^{4-4}\times(-2)\times(-2) \)

\( \qquad\quad = -3 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}1 & 2 & 0 \\ {-}1 & 0 & 1 \\ 0 & 1 & 2\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [0, 1, 2]

\( \qquad\quad = (-1)^{3-2}\times1\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & 2 \\ 0 & 1\end{array}\,\right| + (-1)^{3-3}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & 2 \\ {-}1 & 0\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-2}\times1\times1 + (-1)^{3-3}\times2\times2 \)

\( \qquad\quad = 3 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & 2 \\ 0 & 1\end{array}\,\right| = 1\times1 - 2\times0 = 1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & 2 \\ {-}1 & 0\end{array}\,\right| = 1\times0 - 2\times(-1) = 2 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}2 & 2 & {-}2 \\ {-}1 & 0 & 1 \\ 0 & 1 & 2\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [-2, 1, 2]

\( \qquad\quad = (-1)^{3-1}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & 0 \\ 0 & 1\end{array}\,\right| + (-1)^{3-2}\times1\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 2 \\ 0 & 1\end{array}\,\right| + (-1)^{3-3}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 2 \\ {-}1 & 0\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times(-2)\times(-1) + (-1)^{3-2}\times1\times2 + (-1)^{3-3}\times2\times2 \)

\( \qquad\quad = 4 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}{-}1 & 0 \\ 0 & 1\end{array}\,\right| = (-1)\times1 - 0\times0 = -1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 2 \\ 0 & 1\end{array}\,\right| = 2\times1 - 2\times0 = 2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 2 \\ {-}1 & 0\end{array}\,\right| = 2\times0 - 2\times(-1) = 2 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}2 & 2 & {-}2 \\ 1 & 2 & 0 \\ 0 & 1 & 2\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [-2, 0, 2]

\( \qquad\quad = (-1)^{3-1}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & 2 \\ 0 & 1\end{array}\,\right| + (-1)^{3-3}\times2\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 2 \\ 1 & 2\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times(-2)\times1 + (-1)^{3-3}\times2\times2 \)

\( \qquad\quad = 2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & 2 \\ 0 & 1\end{array}\,\right| = 1\times1 - 2\times0 = 1 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 2 \\ 1 & 2\end{array}\,\right| = 2\times2 - 2\times1 = 2 \)

\( \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}2 & 2 & {-}2 \\ 1 & 2 & 0 \\ {-}1 & 0 & 1\end{array}\,\right| \qquad \) （余因子展開） 第3列で展開する : [-2, 0, 1]

\( \qquad\quad = (-1)^{3-1}\times(-2)\times\left|\,\begin{array}{@{}rwr{15pt}@{}}1 & 2 \\ {-}1 & 0\end{array}\,\right| + (-1)^{3-3}\times1\times\left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 2 \\ 1 & 2\end{array}\,\right| \)

\( \qquad\quad = (-1)^{3-1}\times(-2)\times2 + (-1)^{3-3}\times1\times2 \)

\( \qquad\quad = -2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}1 & 2 \\ {-}1 & 0\end{array}\,\right| = 1\times0 - 2\times(-1) = 2 \)

\( \qquad \left|\,\begin{array}{@{}rwr{15pt}@{}}2 & 2 \\ 1 & 2\end{array}\,\right| = 2\times2 - 2\times1 = 2 \)